How to make particles change colour over time in Eeevee?
I know of two ways of doing it.
A: Add a "time node" using animation nodes, and point in "frame" to some math node then to "color"

B: Use Key Frames on the Color ramp. Click on the first color, set to the frame you want that color to be, in the shader, mouse over the color bar, press I.  Change frame, change color ramp color, mouse over color bar, press I.

NOTE: This changes all the particles color at once.
Note 2: The color ramp, as you have it, won't show up in Eevee, but will in Cycles.

This is similar to: How do I get particles to change color over time?